for Determining the Optimal production and inventory plan to minimize the total cost? Including the fix cost, variable cost, production cost and inventory holding cost over the next $3$ month.

$2)$ Define your decision variables clearly, formulate your model, include objective function and constrains. You do not need to solve it by hand.

$3)$ Use dynamic programming to find the optimal production and inventory plan to minimize the total cost. State the decision clearly. Show all the procedures of the optimal solution.

$4)$ Suppose now that the demand is not known, there is a shortage cost per unit considered, given the demand uncertainty. Describe the methodology in detail for determining the optimal production and inventory plan to minimize the expected total cost